Lineamientos para el establecimiento del proceso de manejo de incidentes de seguridad:

Bulk microphysical parametrizations (e.g., Hsie et al., 1980; Lin et al., 1983; Rutledge and Hobbs, 1983) describing the formation and evaporation of dierent precipitation species are typically formulated such that they are applicable on small scales in regions that are either totally clear or cloudy. Their application in large volumes such as a GCM grid box which can contain both cloudy and cloud free areas is therefore not straightforward. Several approaches to this problem have been proposed. The simplest approach is to ignore partial cloudiness and assume a cloud fraction of one whenever condensate occurs in a grid box (e.g., Fowler et al., 1996). Schemes that parametrize cloud fraction often use in microphysical calculations the in-cloud water content lc=l=a where l is the grid-mean water/ice content and a is the

cloud fraction (Tiedtke, 1993). Until very recently (Bechthold et al., 1993; Tiedtke, 1993; Rotstayn, 1997), the rather obvious fact that partial cloud fraction yields precipitation that only covers a fraction of the grid box has been ignored.

In order to systematically assess the eects of partial cloudiness and partial precipitation coverage a subgrid-scale precipitation representation is developed. The basic idea is to subdivide each grid box intoN sub-columns (N = 20 is used for most of this study) in which the cloud fraction is assigned to be zero or one at every model level. The microphysical parametrization is then applied to each of the sub-columns and the relevant grid-mean quantities (e.g., precipitation ux and evaporation rates) are calculated by summing up the values over all sub-columns. This is equivalent to an increased horizontal resolution for the microphysical calculations. A possible distribution of cloudy and clear sky sub-columns for the distribution of cloud fraction in Figure 6.3 is shown in Figure 6.4. To arrive at the distribution of cloudy and clear-sky sub-columns shown several assumptions were made, the details of which are explained below.

Firstly it is assumed that clouds completely ll the grid box in the vertical; i.e., the fraction of the grid volume that contains cloud is equal to the fraction of the horizontal area of a grid box that contains cloud. Although many clouds have thicknesses less than 500 meters (Wang and Rossow, 1995), this may not be too bad an approximation for the ECMWF model which

96 6. Cloud fraction and microphysics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sub−column number 1 6 11 16 21 26 31 Model level

Figure 6.4: Arrangement of cloudy (black squares) and clear-sky columns for the cloud fraction prole in

Figure 6.3 following the subgrid-scale algorithm.

has 31 to 60 levels in the vertical (see Chapter 3) with typical resolution of 40 hPa (about 400 to 700 m) in the troposphere.

Secondly, at each level the specication of which sub-columns contain cloud is entirely con- sistent with the cloud overlap assumption used for the subgrid-scale ux calculations in the radiation scheme. The overlap assumption currently used in the radiation scheme is that of maximum-random overlap (Geleyn and Hollingsworth, 1979; Morcrette and Fouquart, 1986; Section 3.2). It can be described using the following equation which species the total hor- izontal area, Ck_{, covered by clouds between the top of the atmosphere and a given model}

level k as : 1;Ck 1;Ck ;1 = 1 ;max(ak ;1;ak) 1;min(ak ;1;1 ;) ; (6.1)

where ak _{is the cloud fraction of level k , = 10};6, and k = 1 for the top model level.

C0 and a0 are set to zero. This equation yields random overlap for clouds that do not

occur in adjacent vertical levels but maximum overlap if clouds occur at adjacent levels with cloud fraction monotonically increasing or decreasing with height. This is broadly consistent with the data on cloud overlap of Tian and Curry (1989). At each level, the number of sub-columns that contain cloud is dened to be the nearest integer value of 20ak.

The use of equation (6.1) provides a total cloud cover,Ck_{, by applying it from the model top}

6.2. Cloud and precipitation overlap - The problem 97 ak _{alone is not sucient to unambiguously assign the distribution of cloudy sub-columns. In}

Figure 6.4 for instance it is obvious thatCk _{does not change below model level 11, where it}

reaches a value of 1. Hence, the ve cloudy sub-columns in level 14, where a14

0:25, could

be placed in any sub-box without violating (6.1). Therefore, two additional assumptions are made: i) in the spirit of maximum overlap for clouds in adjacent levels, clouds are assigned to those sub-columns, which contain cloud in the layer immediately above in preference to those sub-columns which do not contain cloud in the layer immediately above, but do have clouds higher in the same sub-column, and ii) the assignment of cloudy boxes begins from the sub- column furthest to the \left" that fullls i). Since especially ii) is rather arbitrary, sensitivity tests to the placement of cloudy sub-columns will be carried out in a later subsection. The \left" assumption is used as the default assumption for the rest of this study because this is the cloud placement that is implicitly assumed in the current scheme (henceforth referred to as the T93 scheme as before).

The amount of liquid water and ice at each level is calculated over the mean cloudy area of the grid box as described in Tiedtke (1993). It must then be divided among the cloudy sub-columns at each level. For want of a better method, each cloudy sub-column is assigned the same amount of liquid water and ice assuming a constant in-cloud water/ice content dened as

lkc;int = _akintlk ; (6.2)

where lk _{is the grid-mean liquid water/ice content and akint is a rounded cloud fraction}

calculated as the fraction of sub-boxes at each level that contain cloud. It is necessary to use a rounded cloud fraction in the denition of lkc;int in order to conserve water. After the allocation of the condensed water in the sub-columns, the same microphysical formulae used for the original model are applied to each sub-column separately. That is, for each sub-box, the generation and evaporation of precipitation is calculated at each level. Averaging over sub-columns yields grid-mean quantities (precipitation and evaporation rates) that can be compared to the T93 parametrization. In all calculations a homogeneous distribution of tem- perature at the beginning of the microphysical calculations is assumed, i.e. each sub-column has the same temperature initially. Through melting and evaporation of precipitation the temperature will change dierently in each sub-column. The new grid-mean temperature is calculated from the grid-mean melting and evaporation rates that are calculated by averaging over their values in the individual sub-columns.

98 6. Cloud fraction and microphysics In applying the microphysical formulae to the sub-columns, two changes are necessary. The rst change is to the melting of snow. In the current model, the amount of melting is limited such that the whole grid box would be cooled back to the freezing temperature over a time scale = 5h, even if precipitation covers only a small fraction of the grid box. For the subgrid-scale precipitation model, the amount of melting is limited such that only the sub- column in which melting occurs can be cooled to the freezing temperature. This implies that if the fraction of the grid box covered by precipitation is less than unity, the energy available for melting is smaller in the subgrid-scale precipitation parametrization than in T93. The main eect of this dierence is to spread melting further in the vertical.

The second change is to the evaporation of precipitation. The formula for evaporation of precipitation (Equation 3.31) depends in part on the humidity of the air into which the precipitation is evaporating. Instead of using the grid-mean humidity in the formula as is done in the current parametrization, an estimate of the humidity of the clear portion of the grid box is calculated using the cloud fraction and the grid-mean humidity. Assuming that the temperature inside the cloud is the same as the grid-mean temperature, the grid-mean specic humidity (qv) is the sum of the saturation value at the grid-mean temperature (qs)

in the cloudy portion of the grid and a mean clear sky value humidity (qvclr):

qv =aqs+ (1;a)qvclr (6.3)

To calculate the evaporation of precipitation in the subgrid-scale precipitation model it is assumed that each cloud-free sub-column has a value of specic humidity equal to the value of qvclr which satises (6.3).

In implementing the subgrid-scale precipitation model, special treatment was given to those grid boxes which have a cloud fraction less than one-half of the size of a sub-column (i.e., ak _{< 0:025 for a model with 20 sub-columns). Normally, the rounding of cloud fraction to}

the nearest sub-column would assign all sub-columns as clear sky and no precipitation could be generated from clouds with these small cloud fractions. In order to avoid problems of that nature, it is required that if the cloud fraction is greater than zero then at least one sub-box in layer k must be lled with cloud. Water conservation in this case is ensured by redistributing the water quantities over the whole sub-box.

It is worthwhile pointing out that the distribution of the cloudy columns in Figure 6.4 should not be interpreted as a spatially contiguous distribution. Shifting all columns randomly, treating each one as a whole in the vertical, will by construction not change the results of

6.2. Cloud and precipitation overlap - The problem 99 the simulation since each column is treated as an independent quantity. It is therefore better to think of the distribution of the cloudy columns as a probability state given the set of rules outlined above.

6.2.2 Comparison of the subgrid-scale precipitation model with